What is the probability that not more than 2 wells are pure?

Question

Eliza Beth13 · Accepted Answer

To find the probability that not more than 2 wells are pure, we need some additional information. Specifically, we need to know the total number of wells and the probability that each individual well is pure. Without this information, it is not possible to provide a specific numerical solution. However, I can explain the general approach to solving such a problem.
Let's assume that we have a total of

n

wells, and the probability of a well being pure is

p

. We want to find the probability that not more than 2 wells are pure.
To solve this, we can consider the cases where 0, 1, or 2 wells are pure, and then sum up the probabilities of these cases.
1. Zero pure wells: The probability that no well is pure can be calculated as

(1 - p)^{n}

. This is because for each individual well, the probability of it not being pure is

(1 - p)

, and since all the wells are independent, we multiply these probabilities together.
2. One pure well: The probability that exactly one well is pure can be calculated as

n p (1 - p)^{n - 1}

. Here, we choose one well out of the

n

wells to be pure, which can be done in

n

ways. The chosen well has a probability

p

of being pure, and the remaining

(n - 1)

wells have a probability

(1 - p)

of not being pure.
3. Two pure wells: The probability that exactly two wells are pure can be calculated as

(\binom{n}{2}) p^{2} (1 - p)^{n - 2}

. Here, we choose two wells out of the

n

wells to be pure, which can be done in

(\binom{n}{2})

ways (using combinations). The chosen wells have a probability

p^{2}

of being pure, and the remaining

(n - 2)

wells have a probability

(1 - p)

of not being pure.
To find the probability that not more than 2 wells are pure, we sum up the probabilities from the above three cases:

P ({not more than 2 pure wells}) = (1 - p)^{n} + n p (1 - p)^{n - 1} + (\binom{n}{2}) p^{2} (1 - p)^{n - 2}

Please note that this is a general formula, and the specific values of

n

and

p

would be needed to obtain a numerical solution.

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